A three domain covariance framework for EEG/MEG data

In this paper we introduce a covariance framework for the analysis of EEG and MEG data that takes into account observed temporal stationarity on small time scales and trial-to-trial variations. We formulate a model for the covariance matrix, which is a Kronecker product of three components that correspond to space, time and epochs/trials, and consider maximum likelihood estimation of the unknown parameter values. An iterative algorithm that finds approximations of the maximum likelihood estimates is proposed. We perform a simulation study to assess the performance of the estimator and investigate the influence of different assumptions about the covariance factors on the estimated covariance matrix and on its components. Apart from that, we illustrate our method on real EEG and MEG data sets. The proposed covariance model is applicable in a variety of cases where spontaneous EEG or MEG acts as source of noise and realistic noise covariance estimates are needed for accurate dipole localization, such as in evoked activity studies, or where the properties of spontaneous EEG or MEG are themselves the topic of interest, such as in combined EEG/fMRI experiments in which the correlation between EEG and fMRI signals is investigated.Comment: 25 pages, 8 figures, 1 tabl

Group Symmetry and non-Gaussian Covariance Estimation

We consider robust covariance estimation with group symmetry constraints. Non-Gaussian covariance estimation, e.g., Tyler scatter estimator and Multivariate Generalized Gaussian distribution methods, usually involve non-convex minimization problems. Recently, it was shown that the underlying principle behind their success is an extended form of convexity over the geodesics in the manifold of positive definite matrices. A modern approach to improve estimation accuracy is to exploit prior knowledge via additional constraints, e.g., restricting the attention to specific classes of covariances which adhere to prior symmetry structures. In this paper, we prove that such group symmetry constraints are also geodesically convex and can therefore be incorporated into various non-Gaussian covariance estimators. Practical examples of such sets include: circulant, persymmetric and complex/quaternion proper structures. We provide a simple numerical technique for finding maximum likelihood estimates under such constraints, and demonstrate their performance advantage using synthetic experiments

Joint Covariance Estimation with Mutual Linear Structure

We consider the problem of joint estimation of structured covariance matrices. Assuming the structure is unknown, estimation is achieved using heterogeneous training sets. Namely, given groups of measurements coming from centered populations with different covariances, our aim is to determine the mutual structure of these covariance matrices and estimate them. Supposing that the covariances span a low dimensional affine subspace in the space of symmetric matrices, we develop a new efficient algorithm discovering the structure and using it to improve the estimation. Our technique is based on the application of principal component analysis in the matrix space. We also derive an upper performance bound of the proposed algorithm in the Gaussian scenario and compare it with the Cramer-Rao lower bound. Numerical simulations are presented to illustrate the performance benefits of the proposed method

A Maximum Entropy solution of the Covariance Extension Problem for Reciprocal Processes

Stationary reciprocal processes defined on a finite interval of the integer line can be seen as a special class of Markov random fields restricted to one dimension. Non stationary reciprocal processes have been extensively studied in the past especially by Jamison, Krener, Levy and co-workers. The specialization of the non-stationary theory to the stationary case, however, does not seem to have been pursued in sufficient depth in the literature. Stationary reciprocal processes (and reciprocal stochastic models) are potentially useful for describing signals which naturally live in a finite region of the time (or space) line. Estimation or identification of these models starting from observed data seems still to be an open problem which can lead to many interesting applications in signal and image processing. In this paper, we discuss a class of reciprocal processes which is the acausal analog of auto-regressive (AR) processes, familiar in control and signal processing. We show that maximum likelihood identification of these processes leads to a covariance extension problem for block-circulant covariance matrices. This generalizes the famous covariance band extension problem for stationary processes on the integer line. As in the usual stationary setting on the integer line, the covariance extension problem turns out to be a basic conceptual and practical step in solving the identification problem. We show that the maximum entropy principle leads to a complete solution of the problem.Comment: 33 pages, to appear in the IEEE Trans. Aut. Cont